**LOGIC
IN SOUTHERN CALIFORNIA
**

Saturday, November 15, 2014

**UCLA Math Sciences 6627**

**Funded
by NSF Grant DMS-1044604**

**Schedule:**

1:30 - 2:20 Jay Williams (Caltech), What descriptive set theory can tell us about group theory.

2:40 - 3:30 Aleksandra Kwiatkowska (UCLA), A generalization of the finite Gowers' Ramsey Theorem with applications to the dynamics of the homeomorphism group of the Lelek fan.

4:00 - 4:50 Spencer Unger (UCLA), The tree property.

5:10 - 6:00 Trevor Wilson (UC-Irvine), Determinacy models and good scales at singular cardinals.

**Abstracts**
**Driving
directions and parking** **Organizers
Previous
meetings**

Jay Williams: **What descriptive set theory can tell us about group theory**.

**Abstract:**
Descriptive set theory gives us tools for analyzing the complexity of groups.
For example, these tools can be used to show that classifying
finitely generated solvable groups up to isomorphism is very hard in
a precise sense. They can also show that sets of groups given by chain
conditions are wild, and this can even be used to prove existence results
in group theory. We will discuss the interplay in these proofs between
constructions from group theory and tools from descriptive set theory.

Aleksandra Kwiatkowska: **A generalization of the finite Gowers' Ramsey Theorem with applications to the dynamics of the homeomorphism group of the Lelek fan**.

**Abstract:**
The Lelek fan is the unique up to homeomorphism compact and connected subspace of the cone over the Cantor set that has a dense set of endpoints.
We generalize a finite version of the Gowers' Ramsey theorem to multiple tetris-like operations and we apply it to show extreme amenability
of a certain subgroup of the homeomorphisms group of the Lelek fan. This is joint work with Dana Bartosova.

Spencer Unger: **The tree property**.

**Abstract:**
We explore best known partial results towards a positive answer to a question
of Magidor: "Is it consistent that every regular cardinal greater than
\(\aleph_1\) has the tree property?" Some recent progress has made a
number of difficult looking open problems more accessible.

Trevor Wilson: **Determinacy models and good scales at singular cardinals**.

**Abstract:**
A typical result in inner model theory might say: if the successor of
a cardinal \(\lambda\) is computed correctly by some inner model, then
some combinatorial incompactness principle holds at \(\lambda\). We
obtain an analogous result for determinacy in place of inner model
theory: if \(\lambda\) is a singular strong limit cardinal and
\(\lambda^+\) is the "definable successor" of \(\lambda\) in the sense of
a generic determinacy model (such as the derived model at \(\lambda\) in
the case where \(\lambda\) is a singular limit of Woodin cardinals),
then there is a good scale at \(\lambda\).

Driving directions and parking:

From the 405 North

- Take 405 (San Diego Freeway) to Wilshire Blvd. East
- Travel east three blocks to Westwood Blvd.
- Turn Left on Westwood Blvd.
- Travel five blocks to Information & Parking Booth

From the 405 South

- Take 405 (San Diego Freeway) to Sunset East
- Take Sunset east to Westwood Plaza
- Turn right on Westwood Plaza
- Proceed straight to Information & Parking Booth

From the east via the 10 (Santa Monica Freeway)

- Take 10 (Santa Monica Freeway) to 405 (San Diego Freeway) North
- Take 405 (San Diego Freeway) to Wilshire Blvd. East
- Travel east three blocks to Westwood Blvd.
- Turn left on Westwood Blvd.
- Travel five blocks to Information & Parking Booth

To park on campus you will need to purchase a daily parking permit at the Information & Parking Booth. The nearest parking lots to the Department of Mathematics are 9, 8, and 2.

Organizers: Alexander Kechris, Itay Neeman, Martin Zeman

**UCLA, May
12, 2012**